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Differential Eq using Substitution help

  1. Sep 13, 2010 #1
    Hey. I am having a hard time solving this problem.


    (x^2)y' + 2xy = 5y^4

    I get as far as simplifying to
    y' = [(5y^4)/(x^2)] - 2y/x

    Then use v: y/x and y: vx & y': v'x + v

    And get

    v'x + v = [5(v^4)(x^2)] - 2v


    And then I get lost. Any help would be appreciated. Thanks!
     
  2. jcsd
  3. Sep 13, 2010 #2
    The convenient substitution is v=1/y^3
     
  4. Sep 15, 2010 #3
    Agreed. while [tex]v = y/x[/tex] does work quite frequently (at least in your homework problems I imagine), you have probably learned about other trademark substitutions of different types of equations. Specifically, this is a Bernoulli type equation (and, it has its own characteristic substitution, and JJacquelin has informed which specific substitution to use in this problem). Learning to diagnose a differential equation and determine what type it is, is a quick way to know immediately how to solve them. I encourage you to look up Bernoulli differential equations online, you will find a lot of material.
     
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